Modelling of Functional Profiles and Explainable Shape Shifts Detection: An Approach Combining the Notion of the Fréchet Mean with the Shape-Invariant Model

Abstract

A modelling framework suitable for detecting shape shifts in functional profiles combining the notion of the Fréchet mean and the concept of deformation models is developed and proposed. The generalized mean sense offered by the Fréchet mean notion is employed to capture the typical pattern of the profiles under study, while the concept of deformation models, and in particular of the shape-invariant model, allows for interpretable parameterizations of the profile’s deviations from the typical shape. The EWMA-type control charts compatible with the functional nature of data and the employed deformation model are built and proposed, exploiting certain shape characteristics of the profiles under study with respect to the generalized mean sense, allowing for the identification of potential shifts concerning the shape and/or the deformation process. Potential shifts in the shape deformation process are further distinguished into significant shifts with respect to amplitude and/or the phase of the profile under study. The proposed modeling and shift detection framework is implemented to a real-world case study, where daily concentration profiles concerning air pollutants from an area in the city of Athens are modeled, while profiles indicating hazardous concentration levels are successfully identified in most cases.

1. Introduction

The task of efficient modelling and the accurate identification of potential shifts for functional profiles has attracted the interest of the statistical community the later years. Such data structures are met more and more in practice in many fields, for instance, in medicine and medical imaging, in electricity management, in water distribution networks, in environmental sciences, etc. The development of more elaborate and complex monitoring schemes is required for the statistical treatment of dynamical and functional data structures in order to reveal and recover important information concerning the special features of the process under study. The resulting functional data (and their increased complexity) is an issue that at a first stage can be handled either by the application of nonlinear models or by non-parametric or semi-parametric approaches (e.g., kernel-based estimators, wavelets, etc.). However, as far as statistical control is concerned, for these functional models, the classical control tools may not be appropriate; partly because of their functional nature and partly because the output of such models may not be appropriately accommodated in a vector space framework. This fact imposes the need for the development of new monitoring mechanisms compatible with the functional data settings. Particularly, in the field of functional profile monitoring, several approaches have been proposed lately, including non-parametric regression or wavelets approaches [1,2,3], interpolation schemes [4,5], PCA and functional PCA-based methods [6,7,8], functional regression schemes [5,9,10], approaches exploiting the intrinsic geometry of appropriate statistical manifolds or relying on the notion of statistical depth [11,12], among others. The majority of the aforementioned approaches attempt to thoroughly modify the modelling procedure, but still rely on the current settings of process control theory and practice since current monitoring tools can better treat the new modelling setup and accommodate certain of the salient features of the data in question (such as functional dependence, etc.).

A crucial issue in studying the functional profiles from the monitoring perspective concerns the actual definition of the typical or the so-called in control (IC) behavior. Standard approaches assume that the under study data can be represented as the points of a finite dimensional Euclidean space (e.g., of ${\mathbb{R}}^d$) of suitable dimensionality related to the number of features under consideration, possibly carrying a correlation structure or displaying variability which is essentially modeled under the assumption of a probability law similar or sufficiently close to a Gaussian one. In the context of functional data, the above setting is not appropriate for a number of reasons. To name just a few:

(a)

The finite dimensionality assumption is no longer valid as the data in question are infinite dimensional, e.g., curves, shapes, and surfaces.

(b)

The dependence structure displayed by data may not be sufficiently modeled within the normality assumption since a more complex type of dependencies may be displayed which cannot be approximated by the Gaussian model.

(c)

The observed data may no longer be understood as elements of a vector space but as elements of a space with nonlinear or convex structure (e.g., elements of a general metric space like covariance matrices, curves of particular form, etc.).

Recently, in [13], a more appropriate framework for dealing with functional data was proposed, employing tools from the statistical shape theory (as can be seen in, e.g., [14,15]). The current work emphasizes the appropriate definition and estimation of the typical behavior of the objects under study through a more general notion of the mean that is applicable for metric space valued data, the Fréchet mean [16,17], combined with the framework of deformation models. The approach that is proposed and developed is quite general and can be applied to any functional object (e.g., curves, surfaces, etc.). However, in order to properly motivate the proposed methodology, the case of profiles represented by curves is studied in this work. Attempting to provide a more concrete framework, the case of the shape-invariant model (SIM) is considered a functional modelling approach which is widely applicable to various contexts. Nevertheless, the discussed framework is extendable to any other deformation model choice beyond SIM.

The purpose of this work is two-fold. First, a modelling framework for representing functional profiles as deformations of a reference typical profile under the shape-invariant model parameterization is proposed. The typical profile is modeled as the Fréchet mean of the sample of the provided IC profiles, while the IC distributions of the deformation features according to the SIM parameterization are induced through a registration step of each IC profile to the typical one. Therefore, at this stage, the typical IC pattern is recovered while acceptable deformations (or deviations) from the typical pattern are expressed in terms of the obtained distributions of the deformation parameters that the SIM incorporates. Secondly, a profile monitoring procedure is built upon these characteristics to detect potential shifts in newly sampled profiles. In particular, employing the adopted deformation modelling approach and the notion of the Fréchet mean, a shift detection scheme is developed based on the rationale of an exponentially weighted moving average (EWMA) monitoring procedure that allows for: (a) the detection of potential shifts in the underlying shape of the profile under study and (b) the detection of potential shifts in the deformation process of the profile compared to the typical pattern. The latter step allows for further investigations of the shift (if one occurred) and categorizing it in terms of shifts in certain characteristics related to amplitude or phase. Both steps of the scheme rely on the relevant deviance processes for the underlying shape and SIM deformations, respectively. Note that the control bounds are determined through the quantiles of the induced empirical IC distributions of each attribute.

The paper is organized in the following way. In Section 2, some important preliminaries concerning the notion of the Fréchet mean and the framework of deformation models are presented, accompanied by a brief literature overview. In Section 3, our modelling setup combining the notion of the Fréchet mean and the shape-invariant model is presented, accompanied by appropriate numerical approximation schemes for estimating the mean pattern (i.e., the reference profile or the Fréchet mean), and registering a sampled profile to the reference one according to the shape-invariant deformation model. Next, in Section 4, the two-step shape shift detection scheme is proposed by appropriately extending the EWMA scheme framework to the case of non-vector spaces. At the end of the same section, the proposed approach is implemented in a real-world example where ambient air quality profiles in an area of the city of Athens are modeled, the proposed shift detection method is applied, and out-of-control behaviors are successfully identified in most of the cases.

2. Literature Overview and Preliminaries

Here, the notion of the Fréchet mean and the framework of the deformation models are briefly introduced. These are the fundamental concepts upon which the present work on functional profiles modelling and monitoring relies.

2.1. Fréchet Mean: A Generalized Version of the Mean

Functional objects like profiles, curves, surfaces, etc. are not objects that necessarily belong to a vector space. For instance, if a profile is subject to some distortion, it is not necessary that this output can be represented under the application of linear operations on the possible profiles of the space under study. In fact, the representation of the distorted profile is not guaranteed by a vector space, even if it is known that the distortion is linear with respect to the amplitude and phase, since the aggregate distortion would be nonlinear with respect to the profile itself (see, for example, the effect of the shape-invariant model which is examined in this work). As such, the typical notion of a mean is not applicable in this case since there is not the typical vector space setting. This motivates the need for a notion of the mean that does not rely on linearity. Such a generalized notion of the mean which is widely applicable is offered by the Fréchet mean [16,18], which for a sample of random elements $\{x_1,\dots ,x_n\}\subset \mathcal{M}$ is defined as

$$x_F:=arg\underset{z\in \mathcal{M}}{min}\mathsf{\Psi }\left(z\right)$$

(1)

where $\mathsf{\Psi }\left(z\right)={\sum }_{j=1}^n{w}_j{d}^2(z,x_j)$ is the generalized Fréchet function of the sample, d is a suitable metric for the space $\mathcal{M}$, and $\{w_1,\dots ,w_n\}$ is a suitable set of weights for the data (the case $w_i=1/n$ corresponds to the standard definition of the Fréchet mean). The minimum value of $\mathsf{\Psi }$ (achieved at $x_F$) corresponds to the variance of the sample. The Fréchet mean is a variational concept that can be obtained using techniques from the calculus of variations and optimization (as can be seen, e.g., in [19]) and its theoretical properties are well documented (see, e.g., [17,20,21,22]). It has been recently used by the statistics community as an important tool in the study of data which cannot be conveniently described as the elements of vector spaces (see, e.g., [23,24,25], etc.). In the monitoring framework presented in this work, the Fréchet mean is employed to capture the typical pattern or mean behavior of an IC sample of functional profiles. However, the performance of this approach significantly depends on the suitability of the deformation model that is chosen to model the variations around the typical pattern (in the case that a semi-parametric approach is adopted) and, of course, the metric sense under which the operations are derived.

2.2. The Framework of Deformation Models

A very natural setting for the study of functional profiles is that of deformation models. Deformation models produce metric space valued random elements which belong to some set $\mathcal{M}=\{f\circ T:T\in V\}$, where f is a deterministic function (often to be specified) characterizing the typical (average) shape, and $T(·)=T(·,\omega )$ is a random deformation typically chosen from a vector space V so that the composition $f\circ T(·,\omega )$ generates a random element from $\mathcal{M}$, satisfying certain qualitative features. Hence, each random element is parameterized in terms of the realization of $T\in V$. For example, f can be the curve modelling the distribution of some pollutant over the day, accounting for patterns like the mean variability of traffic or the mean variability of temperature, while T accounts for specific features that may randomly happen on a particular day and may not repeat. Note that, in many cases of interest, $\mathcal{M}$ is not endowed with a linear structure (e.g., in the case of s-shaped curves) but is rather a subset of a more general metric space. For instance, consider curves of a specific shape as those that appear in medicine (e.g., intra-day blood pressure curves) or those describing the intra-day demand of electric energy in a city or daily consumption patterns over a water distribution network. Such objects do not necessarily belong to a vector space, since summing the two sampled objects does not necessarily lead to an object of the underlying space. This type of statistical modelling has found several applications in practice so far, e.g., in image and signal processing [26,27], in analyzing point processes [28], in medicine [29], in electric energy prediction [30], etc. Restricting attention to the case where $f:I\to \mathbb{R}$ is a function from some interval of I ($I=[0,1]$ without loss of generality) is a suitable choice for the study of nonlinear profiles. A more generic form of a deformation model has been studied in [28], separately modelling amplitude and phase deformation characteristics. Under the perspective of the general amplitude-phase deformation model, the shape relation between two curves f and g is expressed as

$$f\left(t\right)=A\left(g\left({\mathsf{\Phi }}^{-1}\left(t\right)\right)\right)+ϵ\left(t\right)$$

(2)

where $\mathsf{\Phi }:I\to I$ for $I\subset \mathbb{R}$ represents the phase deformation process (its inverse ${\mathsf{\Phi }}^{-1}$ is often referred to as the time-warping function according to the terminology of [31]) and $A:L^2\left(I\right)\to L^2\left(I\right)$ represents the amplitude deformation process, where $L^2\left(I\right)$ denotes the Lebesgue space of the square-integrable functions with domain I. Moreover, $ϵ\left(t\right)$ denotes the relevant error term, i.e., the aspects of curve f that cannot be efficiently captured through a deformation of g under the particular shape deformation modelling approach. Parametric forms of $\mathsf{\Phi }$ and A are conceivable and practical and hence are often used in practice (as can be seen, e.g., in [31,32,33]). Considering affine parametric forms for both the phase and amplitude deformation process, i.e., $A\left(z\right)=\alpha z+b$ and $\mathsf{\Phi }\left(s\right)=\kappa s+\zeta$, reduces the amplitude-phase deformation model to the well-celebrated shape-invariant model on which the presented approach relies.

3. A Functional Modelling Approach Combining the Shape Invariant Model Framework and the Notion of the Fréchet Mean

In this section, a modelling framework for functional profiles is provided which allows for the parameterization of an individual’s profile deviance with respect to the typical pattern. In particular, special attention is given to the shape-invariant deformation model under which linear deformations are assumed with respect to the amplitude and phase characteristics separately (but still nonlinear with respect to the reference shape). Relying on this deformation model, the Fréchet mean of a set of profiles is approximated by appropriately formulating the Fréchet variance the determination problem and proposing a numerical approximation scheme for this purpose. Moreover, the registration problem of a sampled profile to the reference shape (Fréchet mean) is considered, providing the related well-posedness results.

3.1. A Brief Presentation of the Shape-Invariant Deformation Model

The shape-invariant model has been repeatedly discussed and appreciated in the functional modelling literature (as can be seen in, e.g., [32,33,34,35]) due to its applicability in various different contexts and the interpretability of its deformation parameters. Under this setting, an object $f_j$ is modeled as a distortion of the typical profile $f_0$ through the relation

$$f_j\left(t\right)={\beta }_j+{\alpha }_j{f}_0\left(\frac{t-{\zeta }_j}{{\kappa }_j}\right)+{ϵ}_j\left(t\right),t\in I$$

(3)

which can be realized as a special case of the general amplitude-phase deformation model (see, e.g., [28]) considering $A_j\left(z\right)={\alpha }_{j}z+{\beta }_j$ and ${\mathsf{\Phi }}_j\left(s\right)={\kappa }_{j}s+{\zeta }_j$ with $({\alpha }_j,{\beta }_j)$ denoting the deformation parameters related to the scale and location of the amplitude deformation process, while $({\kappa }_j,{\zeta }_j)$ denotes the parameters affecting the scale and location of the phase deformation process (please see Figure 1 for an illustration of the deformation model’s characteristics). In this perspective, the space of profiles $\mathcal{M}$ is identified by the following subset of $L^2$

$$\mathcal{M}:=\left\{f\in L^2:\exists {(\alpha ,\beta ,\kappa ,\zeta )}^{\prime }\in \mathsf{\Theta }\subset {\mathbb{R}}^4\mathrm{where}f\left(t\right)=\beta +\alpha f_0\left(\frac{t-\zeta }{\kappa }\right),t\in I\right\}$$

(4)

with the set $\mathsf{\Theta }$ including the feasible values for the deformation parameters, leading to a metric space setting which does not carry the linear structure of a vector space.

3.2. Determination of the Typical Profile under the Shape-Invariant Model

Assume that there is a training set of profiles $\mathcal{F}:=\{f_1,f_2,\dots ,f_n\}$, and that each profile $f_j$ can be sufficiently modeled as a deformation of the (unknown yet) mean profile $f_0$, as represented in (3), with deformation parameters ${\theta }_j={({\alpha }_j,{\beta }_{,}{\kappa }_j,{\zeta }_j)}^{\prime }$. The amplitude distortions of $f_j$ compared to the reference (Fréchet mean) profile $f_0$ are captured by the deformation function $A(t;{\gamma }_j)={\beta }_j+{\alpha }_j{f}_0\left(t\right)$ with ${\gamma }_j:={({\beta }_j,{\alpha }_j)}^{\prime }$ denoting the amplitude deformation parameters, while phase distortions are captured by the function $\mathsf{\Phi }(t;{\xi }_j)={\zeta }_j+{\kappa }_{j}t$ with ${\xi }_j={({\zeta }_j,{\kappa }_j)}^{\prime }$ denoting the relevant phase deformation parameters. These parameterized effects cannot be specified unless the Fréchet mean is estimated. Following the current modelling approach, a semi-parametric expression of the mean is obtained through averaging. Using the model (3), one obtains a motivation for the definition of the Fréchet mean using the functional data $f_j$. In particular, one may express

$$f_0\left(t\right)={\widehat{f}}_{0,j}(t;{\theta }_j)+{\eta }_j\left(t\right)={\alpha }_j^{-1}\left(f_j({\kappa }_{j}t+{\zeta }_j)-{\beta }_j\right)+{\eta }_j\left(t\right)$$

(5)

where ${\eta }_j(·)$ denotes the distorted error process related to the estimation error relying on the observation from $f_j$. The mean profile $f_0$ must be chosen in such a manner that satisfies the barycenter property, i.e., being simultaneously so close and so far from all the profiles contained in the set $\mathcal{F}$. A standard requirement that has to be met is that the average of the residuals ${\eta }_j$ among all elements, for each time instant $t\in I$, must be zero, i.e., $\frac{1}{n}{\sum }_{j=1}^n{\eta }_j\left(t\right)=0$. As a result, by properly averaging every possible model of $f_0$ as a deformation of each $f_j\in \mathcal{F}$, the following semi-parametric expression is obtained

$${\widehat{f}}_0(t;\mathit{\theta })=\frac{1}{n}\sum _{j=1}^n{\widehat{f}}_{0,j}(t;{\theta }_j)=\frac{1}{n}\sum _{j=1}^n{\alpha }_j^{-1}\left(f_j({\kappa }_{j}t+{\zeta }_j)-{\beta }_j\right)$$

(6)

where $\mathit{\theta }={({\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{\prime },{\mathit{\kappa }}^{\prime },{\mathit{\zeta }}^{\prime })}^{\prime }$ with $\mathit{\alpha }={({\alpha }_1,\dots ,{\alpha }_n)}^{\prime }$, $\mathit{\beta }={({\beta }_1,\dots ,{\beta }_n)}^{\prime }$, $\mathit{\kappa }={({\kappa }_1,\dots ,{\kappa }_n)}^{\prime }$ and $\mathit{\zeta }={({\zeta }_1,\dots ,{\zeta }_n)}^{\prime }$. Clearly, what is represented in (5) is not the pure Fréchet mean, but rather its best approximation under the shape-invariant model. The optimality criterion which is employed for the selection of $\mathit{\theta }$ depends on the metric sense $d_M$ under which the mean element is derived. Recalling the Fréchet function defined in (1), the optimal ${\mathit{\theta }}_{\ast }\in \mathsf{\Theta }$, where $\mathsf{\Theta }$ denotes the feasible set of the deformation parameters, is obtained as the solution to the minimization problem

$$\underset{\mathit{\theta }\in \mathsf{\Theta }}{min}V\left(\mathit{\theta }\right):=\underset{\mathit{\theta }\in \mathsf{\Theta }}{min}\frac{1}{n}\sum _{j=1}^n{d}_M^2({\widehat{f}}_0\left(\mathit{\theta }\right),f_j)$$

(7)

leading to the determination of the Fréchet mean under the current deformation family of models. In general, the required centrality properties for the amplitude and phase deformation functions are

• $A_1\circ A_2\circ \cdots \circ A_n=Id$
• ${\mathsf{\Phi }}_1\circ {\mathsf{\Phi }}_2\circ \cdots \circ {\mathsf{\Phi }}_n=Id$.

The conditions for the case of the shape-invariant model are equivalently stated by the set of constraints for the related deformation parameters

$$\prod _{j=1}^n{\alpha }_j=1,\sum _{j=1}^n{\beta }_j=0,\prod _{j=1}^n{\kappa }_j=1,\sum _{j=1}^n{\zeta }_j=0$$

(8)

with ${\alpha }_j,{\kappa }_j$ being strictly positive and upper-bounded by some ${\alpha }_{max},{\kappa }_{max}$, while ${\beta }_j$ and ${\zeta }_j$ are constrained to some appropriate intervals $[{\beta }_{min},{\beta }_{max}]$ and $[{\zeta }_{min},{\zeta }_{max}]$, respectively (for all $j=1,2,\dots ,n$). These properties ensure that the total deformation effect sums up to zero, and that the derived minimizer from (7) represents the optimal reference profile since all elements in the training set are aligned with respect to this profile, either in terms of amplitude or phase distortions. To make these conditions clearer, consider for instance the simple case of two profiles. Under the aforementioned centrality conditions, the aggregate amplitude deformation effect is required to satisfy

$$\begin{array}{c}\hfill (A_1\circ A_2)\left(g\right)=g\iff {\beta }_1+{\beta }_2+{\alpha }_1{\alpha }_{2}g=g,\forall g\in L^2\end{array}$$

resulting in ${\beta }_1+{\beta }_2=0$ and ${\alpha }_1{\alpha }_2=1$; while concerning the aggregate phase deformation effect, it is required that

$$\begin{array}{c}\hfill ({\mathsf{\Phi }}_1\circ {\mathsf{\Phi }}_2)\left(t\right)=t\iff {\zeta }_1+{\zeta }_2+{\kappa }_1{\kappa }_{2}t=t,\forall t\in I\end{array}$$

resulting in ${\zeta }_1+{\zeta }_2=0$ and ${\kappa }_1{\kappa }_2=1$. Then, the obtained mean element provides a comparison pattern with respect to which all individual elements in the training set are aligned to in terms of both phase and amplitude.

Remark 1.

Note that a potential minimizer ${\mathit{\theta }}_{\ast }$ of (7) contains the optimal deformation parameters concerning the approximation of the Fréchet mean of the set $\mathcal{F}$ under the SIM approach. For a specific profile $f_j\in \mathcal{F}$, the related deformation parameters ${\theta }_{\ast ,j}$ contained in the vector ${\mathit{\theta }}_{\ast }$ does not necessarily coincide with the parameter choices that best represent the current profile as an SIM deformation with respect to the mean shape. This happens since the optimal solution of (7) is subject to the set of constraints stated in (8) which are much more restrictive than those that are required for registering some specific $f_j$ to the mean shape (this matter is discussed separately in the following sections). Clearly, ${\mathit{\theta }}_{\ast }$ is the choice of parameters indicating the optimal position (or tuning) of the semi-parametric mean shape model stated in (6) with respect to all the elements in the set $\mathcal{F}$. Clearly, this is not necessarily the same as collecting the optimal deformations of each $f_j\in \mathcal{F}$ with respect to the (estimated) mean shape ${\widehat{f}}_0\left(\mathit{\theta }\right)$.

The problem of estimating the mean profile (typical shape) under the SIM framework and employing the $L^2$ metric sense (i.e., $d_M(f,g)={\parallel f-g\parallel }_2$ for any $f,g\in \mathcal{M}$) is now considered. The resulting Fréchet function minimization problem is characterized by the objective function

$$V\left(\mathit{\theta }\right):=\frac{1}{n}\sum _{j=1}^n\langle f_j-{\widehat{f}}_0\left(\mathit{\theta }\right),f_j-{\widehat{f}}_0\left(\mathit{\theta }\right)\rangle$$

(9)

where $\langle f,g\rangle :={\int }_{I}f\left(s\right)g\left(s\right)ds$ denotes the inner-product in the $L^2\left(I\right)$ for any $f,g\in L^2\left(I\right)$ and ${\widehat{f}}_0\left(\mathit{\theta }\right)$ as stated in (6). Following the discussion in the previous section, the problem of minimizing (9) over the set $\mathsf{\Theta }$, denoting the feasible set for $\mathit{\theta }$ satisfying all the constraints stated in (8), is studied. In order to guarantee the uniqueness of the solution (existence is guaranteed by the continuity with respect to $\mathit{\theta }$), appropriate regularizations terms are necessary. For this reason, the quadratic regularization term $R_{\rho }\left(\mathit{\theta }\right)=\frac{1}{2\rho }{\parallel \mathit{\theta }\parallel }_2^2$ is introduced, driven by the sensitivity parameter $\rho >0$. This regularization transforms, for a certain value of $\rho$ and below, the problem into a strictly convex one. Besides the convexity-correction of the problem, the regularization term enhances the stability of the problem, not allowing each deformation parameter to exceed a certain threshold value, which in combination with the constraints provided by the set $\mathsf{\Theta }$, reduces the effect of potential extreme profiles in $\mathcal{F}$ to the estimation of the mean, making the estimation procedure more robust. Modifying the problem in this way, when more than one minimizer exists, a selection criterion among the potential solutions is introduced, setting a preference order that promotes those that cause less dramatic deformations with respect to the original profiles in $\mathcal{F}$. The related results follows (for the relevant proof please, as can be seen in Appendix A.1).

Proposition 1.

There exists ${\rho }_0>0$ for which the regularized Fréchet variance determination problem

$$\underset{\mathit{\theta }\in \mathsf{\Theta }}{min}\left\{V\left(\mathit{\theta }\right)+\frac{1}{2\rho }{\parallel \mathit{\theta }\parallel }_2^2\right\}$$

(10)

admits a unique solution for any $\rho \in (0,{\rho }_0]$.

3.3. A Numerical Spitting Approach for the Approximation of the Mean Profile

Next, the development of a numerical approximation scheme for estimating the mean profile under the SIM framework is considered, i.e., approximating the optimal solution of the problem (7). First, the dependence of function V is distinguished by the amplitude deformation parameters $\mathit{\gamma }={({\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{\prime })}^{\prime }$ and to the phase deformation parameters $\mathbf{\xi }={({\mathit{\kappa }}^{\prime },{\mathit{\zeta }}^{\prime })}^{\prime }$. Splitting the parameters into two distinct groups is a key step towards the development of an efficient numerical approximation scheme. Ignoring at first the centrality constraints (8) and under the appropriate transformations of the parameters contained in $\mathit{\gamma }$ (considering $\mathit{\gamma }:={({\overline{\mathit{\alpha }}}^{\prime },{\overline{\mathit{\beta }}}^{\prime })}^{\prime }$ with ${\overline{\alpha }}_j:=1/{\alpha }_j$ and ${\overline{\beta }}_j:={\beta }_j{\overline{\alpha }}_j$ for $j=1,2,\dots ,n$), the function V quadratically depends on the amplitude deformation parameters, while as far as the parameter vector $\mathbf{\xi }$ is concerned, the dependence is non-quadratic and importantly non-convex (at least in general cases). Directly treating the problem with respect to all the parameters results in a non-convex and computationally expensive minimization problem which is difficult to treat. Alternatively, an iterative parameter-splitting minimization scheme is proposed, which treats the problem in two separate stages and takes advantage over the quadratic dependence of V with respect to $\mathit{\gamma }$ and by convexly approximating the solution with respect to $\mathbf{\xi }$. The Tikhonov-type regularizations are considered to both parts of the problem (for different reasons), and through successive projections of the obtained solutions to the constrained sets, an optimal solution is reached.

The space $\mathsf{\Theta }$ is decomposed into two components ${\mathsf{\Theta }}_{\mathsf{\Lambda }}$ and ${\mathsf{\Theta }}_{\mathsf{\Pi }}$ (satisfying $\mathsf{\Theta }={\mathsf{\Theta }}_{\mathsf{\Lambda }}\times {\mathsf{\Theta }}_{\mathsf{\Pi }}$), as follows

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_{\mathsf{\Lambda }}& =& \left\{\mathit{\gamma }={({\overline{\mathit{\alpha }}}^{\prime },{\overline{\mathit{\beta }}}^{\prime })}^{\prime }\in {\mathbb{R}}^{2n}:\prod _{j=1}^n{\overline{\alpha }}_j=1,\sum _{j=1}^n\frac{{\overline{\beta }}_j}{{\overline{\alpha }}_j}=0,{\overline{\alpha }}_j>0,\forall j\right\}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_{\mathsf{\Pi }}& =& \left\{\mathbf{\xi }={({\mathit{\kappa }}^{\prime },{\mathit{\zeta }}^{\prime })}^{\prime }\in {\mathbb{R}}^{2n}:\prod _{j=1}^n{\kappa }_j=1,\sum _{j=1}^n{\zeta }_j=0,{\zeta }_j\in \left[{\zeta }_{min},{\zeta }_{max}\right],\forall j\right\}.\hfill \end{array}$$

(12)

To clarify the separate dependence of V from the amplitude deformation parameters $\mathit{\gamma }$ and from the phase deformation parameters $\mathbf{\xi }$, it is denoted equivalently from now on by the respective Fréchet function $V(\mathit{\gamma },\mathbf{\xi })$. The regularization terms of the form suggested in Proposition 1 are introduced to the problem separately for $\mathit{\gamma }$ and $\mathbf{\xi }$. Fixing a choice $\mathbf{\xi }$, the (unconstrained) minimization problem with respect to $\mathit{\gamma }$ can be solved in closed form, while a regularization term is required for stabilizing the estimation procedure. Then, the solution is projected to its constrained set ${\mathsf{\Theta }}_{\mathsf{\Lambda }}$. Subsequently, fixing the obtained $\gamma$, the (unconstrained) minimization problem with respect to $\mathbf{\xi }$ is solved, introducing a regularization term that will guarantee the convexity of the objective function. The obtained solution $\mathbf{\xi }$ is projected to its constraint set ${\mathsf{\Theta }}_{\mathsf{\Pi }}$ and the above steps are repeated until convergence. The scheme is described in algorithmic formulation below.

The aforementioned alternative projections-splitting scheme converges to a unique solution from standard arguments from the theory of alternating projections. The result is stated in the following proposition (for more details please see Appendix A.2).

Proposition 2.

The following hold:

a. 

Given ${\mathit{\gamma }}^{\left(0\right)}\in {\mathsf{\Theta }}_{\mathsf{\Lambda }}$ and ${\mathbf{\xi }}^{\left(0\right)}\in {\mathsf{\Theta }}_{\mathsf{\Pi }}$, there exists ${\rho }^{\ast }>0$ for which the optimization problems (13) and (14) admit unique solutions for any $\rho \in (0,{\rho }^{\ast }]$.

b. 

If ρ belongs to $(0,{\rho }^{\ast }]$, then the iterative spitting-projection numerical scheme stated in Algorithm 1 converges to a unique solution.

Algorithm 1: Iterative splitting-projection scheme for Fréchet mean approximation

0. Set$k=1$, ${\mathit{\gamma }}^{\left(0\right)}={(1,1,\dots ,1,0,0,\dots ,0)}^{\prime }$, ${\mathbf{\xi }}^{\left(0\right)}={(1,1,\dots ,1,0,0,\dots ,0)}^{\prime }$and specify a tolerance level ($ϵ>0$). 1. Solve the amplitude-unconstrained optimization problem $$\widehat{\mathit{\gamma }}=arg\underset{\mathit{\gamma }\in {\mathbb{R}}^{2n}}{min}\left\{V(\mathit{\gamma },{\mathbf{\xi }}^{(k-1)})+\frac{1}{2\rho }\left({\parallel \mathit{\gamma }\parallel }_2^2\right\}+{\parallel {\mathbf{\xi }}^{(k-1)}\parallel }_2^2\right),\rho >0$$ (13) and then compute the projection${\mathit{\gamma }}^{\left(k\right)}=Pro{j}_{{\mathsf{\Theta }}_{\mathsf{\Lambda }}}\left(\widehat{\mathit{\gamma }}\right)$ 2. Solve the phase-unconstrained optimization problem $$\widehat{\mathbf{\xi }}=arg\underset{\mathbf{\xi }\in {\mathbb{R}}^{2n}}{min}\left\{V({\mathit{\gamma }}^{\left(k\right)},\mathbf{\xi })+\frac{1}{2\rho }\left({\parallel \mathbf{\xi }\parallel }_2^2+{\parallel {\mathit{\gamma }}^{\left(k\right)}\parallel }_2^2\right)\right\},\rho >0$$ (14) and then compute the projection${\mathbf{\xi }}^{\left(k\right)}=Pro{j}_{{\mathsf{\Theta }}_{\mathsf{\Pi }}}\left(\widehat{\mathbf{\xi }}\right)$ 3. If $\parallel {\mathit{\gamma }}^{\left(k\right)}-{\mathit{\gamma }}^{(k-1)}\parallel <ϵ$and$\parallel {\mathbf{\xi }}^{\left(k\right)}-{\mathbf{\xi }}^{(k-1)}\parallel <ϵ$stop and set$({\mathit{\gamma }}_{\ast },{\mathbf{\xi }}_{\ast })=({\mathit{\gamma }}^{\left(k\right)},{\mathbf{\xi }}^{\left(k\right)})$. Else set$k=k+1$and return to Step 1.

Remark 2.

Note that the solution of Problem (10) can be obtained by Algorithm 1 for appropriate choices of the individual regularization parameters ${\rho }_{\mathsf{\Lambda }}$ and ${\rho }_{\mathsf{\Pi }}$.

3.4. Profile Registration to the Mean Shape

Next, the registration problem of a profile $f_j$ (not necessarily belonging to the training set $\mathcal{F}$) to the mean profile $f_0$ is considered. This can also be realized as the best approximation problem of a $f_j$ through the SIM deformation of $f_0$ (as estimated through the procedure presented in Section 3.3). This leads to the profile registration problem

$$\underset{\theta \in {\mathsf{\Theta }}_R}{min}{V}_j\left(\theta \right)=\underset{\theta \in {\mathsf{\Theta }}_R}{min}\langle f_j-\widehat{f}\left(\theta \right),f_j-\widehat{f}\left(\theta \right)\rangle$$

(15)

where $\widehat{f}\left(\theta \right)=\beta +\alpha f_0\left({\kappa }^{-1}(t-\zeta )\right)$ and

$${\mathsf{\Theta }}_R:=\left\{\theta ={(\alpha ,\beta ,\kappa ,\zeta )}^{\prime }:\begin{array}{cc}\alpha \in (0,{\alpha }_{max}],\hfill & \beta \in [{\beta }_{min},{\beta }_{max}],\hfill \\ \kappa \in (0,{\kappa }_{max}],\hfill & \zeta \in [{\zeta }_{min},{\zeta }_{max}]\hfill \end{array}\right\}$$

(16)

where the lower and upper bounds of the parameters are subject to the characteristics of the profiles $f_0$ and $f_j$, respectively. Although problem (15) admits solutions by standard continuity arguments, it is not necessary for the solution to be unique. This problem practically concerns only the phase deformation parameters $(\kappa ,\zeta )$, since the same problem is strictly convex (quadratic) with respect to the amplitude deformation parameters $(\alpha ,\beta )$. Therefore, a regularization term is needed for the successful treatment of the problem which will importantly play the role of a selection criterion when more than one minimizer exists. In that case, one would like to choose that value for $\theta$ which causes the smallest possible deformation to the reference shape $f_0$ in order to represent $f_j$. In the same spirit with Section 3.3, quadratic regularization terms are incorporated for the appropriate tuning of the aggregate deformation effect, caused by the choice of an arbitrary $\theta ={(\alpha ,\beta ,\kappa ,\zeta )}^{\prime }$, favoring the minimum possible deformation. Then, it can be shown that, for a certain value of the sensitivity parameter $\rho$ and below, the single profile registration problem admits a unique solution. However, note that the introduced regularization term does not necessarily involve all the parameters in $\theta$, but only $\kappa ,\zeta$, since closed-form solutions for $\alpha ,\beta$ can be attained given $\kappa ,\zeta$ (please see the relevant discussion in the Appendix A.3).

Proposition 3.

Let $f_0\in L^2\left(I\right)$ be a reference shape and $f_j\in L^2\left(I\right)$ be a profile to be registered to it as an SIM deformation. There exists ${\rho }_{\ast }>0$ for which the single profile registration problem

$$\underset{\theta \in {\mathsf{\Theta }}_R}{min}\left\{V_j\left(\theta \right)+\frac{1}{2\rho }{\parallel \theta \parallel }_2^2\right\}$$

(17)

where $V_j$ is determined in (15), admitting a unique solution for any $\rho \in (0,{\rho }_{\ast }]$.

For the proof please, see Appendix A.3.

4. The Exponentially Weighted Fréchet Moving Average Scheme for Functional Profiles

Following the characterization and estimation of the typical profile in Section 3, the construction of an EWMA-type procedure for detecting potential shifts with respect to the underlying shape and deformations from it is proposed. The main focus is given on the case of curves representing the evolution of a quantity in a specified time (or space) interval $t\in I$, the pattern of which can be efficiently calibrated by the shape-invariant model as discussed above. The rationale behind the developed control chart, is similar to the classical EWMA charts procedure either for monitoring the mean or the variability process of the statistical quantities under study (see, e.g., [36]).

The novelty of the proposed shift detection scheme relies on the fact that a generalized mean sense is directly incorporated in the chart element estimation procedure, providing an attractive framework for analyzing functional data parameterizing certain characteristics of them. In what follows, a modification of the EWMA monitoring approach is presented, in which the mean sense derived by the notion of the Fréchet mean is deployed for the development of an exponentially weighted scheme, which is able to be directly implemented into the metric space of the profiles under study. This modification enables the development of the first branch of the proposed shift detection scheme, which allows for the monitoring of significant changes to the underlying shape of the profile under study. At the second step, a Fréchet mean-based scheme is built for the surveillance of the shape deformation processes (according to the shape-invariant model) which quantify the deviations of the profile under study from the reference pattern. The aforementioned steps provide a two-stage shift detection framework capable of explaining the potential shifts of the profile under study and interpret them either as shape shifts (i.e., significant shifts in the underlying shape) or deformation shifts (i.e., significant shifts in the aggregate deformation effect with respect to the reference shape). In the sequel, the term exponentially weighted Fréchet moving average (EWFMA) is used when referring to the proposed scheme.

4.1. Shift Detections on the Underlying Shape

Given a training set of n in control (IC) profiles $\mathcal{F}=\{f_1,\dots ,f_n\}$, the typical profile $f_0$ is estimated through the procedure described in Section 3 under the assumption that any IC element is acceptably deviant from the mean shape $f_0$ with respect to the shape-invariant model. Being acceptably deviant means both that (a) the underlying shape of the profile and (b) the respective amplitude-phase deformation process (as captured by the shape-invariant model), produce aggregate deviance levels that do not exceed those ones observed from the IC dataset $\mathcal{F}$. Given the estimated IC behavior $f_0$, for a newly sampled $f_j$, the best approximation according to model (3) is obtained through the solution of the related registration problem stated in (17).

At the first stage of the EWFMA scheme, a chart for monitoring the shape deviance process is constructed, i.e., the deviance of the underlying shape for a profile $f_j$ (after removing the amplitude and phase deformation characteristics as estimated from (17)) from the reference one. The retrieved underlying shape for the j-th profile given the obtained optimal deformation parameters ${\theta }_j$ is defined as

$${\widehat{f}}_{0,j}={\alpha }_j^{-1}\left(f_j({\kappa }_{j}t+{\zeta }_j)-{\beta }_j\right).$$

(18)

Then, the underlying shape deviance between the j-th individual’s underlying shape and the typical shape $f_0$ is calculated as

$$D_j^s:=d_M^2({\widehat{f}}_{0,j},f_0)$$

(19)

where $d_M$ denotes the metric sense under which the deviance is determined ($L^2$-distance in this paper). If the underlying shape for a profile is dramatically divergent from the typical one, then the shape-invariant model is incapable of modelling the current observation and therefore the examined profile should be considered an OOC one (i.e., not explainable from the current deformation model). As a result, the following EWMA-type scheme for monitoring the underlying shape deviance process is proposed:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{D}}_j^s=\lambda D_j^s+(1-\lambda ){\tilde{D}}_{j-1}^s,\lambda \in (0,1],j=1,2,\dots \hfill \\ {\tilde{D}}_0^s=0\hfill \end{array}\right.\end{array}$$

(20)

The bounds beyond which the deviance process provides a shift can be estimated by the empirical distribution of shape deviances from the IC profiles given certain confidence levels.

4.2. Shift Detections on the Deformation Features

Assuming that no shift in the underlying shape is detected, the profile under study can be described by the shape-invariant model, while at the next step, its deformation characteristics (amplitude and phase deformations) are examined. For this task, a separate exponentially weighted chart was developed, relying on the formulation of the shape-invariant model combined with the notion of the Fréchet mean. The resulting chart concerns the deformations deviance process, i.e., the total deformation effect performed by both the amplitude and phase deformations to the underlying shape. In this view, each profile (since it has been already considered as in control in terms of preserving the typical shape $f_0$) is identified by its modelling counterpart, i.e., ${\widehat{f}}_j=\widehat{f}\left({\theta }_j\right)$ where ${\theta }_j$ is the minimizer of the related registration problem stated in (17) and $\widehat{f}\left(\theta \right)$ representing the fitted part of $f_j$ from (3) (without the error term). This second stage of the EWFMA scheme is built directly on the modeled profiles with respect to the Fréchet mean sense. Initializing the monitoring procedure with the reference profile $f_0$, each new element of the scheme is naturally derived as the Fréchet mean of the profiles $f_0$ and ${\widehat{f}}_j$ with the corresponding weights $\lambda$ and $1-\lambda$, leading to the variational problem

$${\tilde{f}}_j:=arg\underset{g\in \mathcal{M}}{min}\left\{\lambda d_M^2(g,{\widehat{f}}_j)+(1-\lambda )d_M^2(g,{\tilde{f}}_{j-1})\right\},{\tilde{f}}_0=f_0.$$

(21)

Since the study of the deformation characteristics with respect to $f_0$ is of particular interest in this work, a reduced version of the variational problem stated above is considered, by substituting $g=\widehat{f}\left(\theta \right)$ resulting in the simpler fitting problem

$$\underset{\theta \in {\mathsf{\Theta }}_R}{min}{V}_{\lambda }\left(\theta \right)=\underset{\theta \in {\mathsf{\Theta }}_R}{min}\left\{\lambda \parallel \widehat{f}\left(\theta \right)-{\widehat{f}}_j{\parallel }_2^2+(1-\lambda ){\parallel \widehat{f}\left(\theta \right)-{\tilde{f}}_{j-1}\parallel }_2^2\right\},\lambda \in (0,1]$$

(22)

where ${\mathsf{\Theta }}_R$ is the set defined in (16). For the same reasons discussed in Section 3.4, the above problem does not necessarily admit a unique solution, and a regularized version of the problem similar to the one in Proposition 3 is considered. In a similar manner, the regularization term turns the problem into a strictly convex one with respect to $\theta$, and at the same time, introduces a selection criterion when multiple minima occur. The related result follows (for details on the proof, please see Appendix B).

Proposition 4.

Let $f_0\in L^2\left(I\right)$ be the mean pattern and $\widehat{f},\tilde{f}\in L^2\left(I\right)$ be two reference profiles. There exists ${\tilde{\rho }}_{\ast }>0$ for which the chart element estimation problem

$$\underset{\theta \in \tilde{\mathsf{\Theta }}}{min}\left\{V_{\lambda }\left(\theta \right)+\frac{1}{2\rho }{\parallel \theta \parallel }_2^2\right\}$$

(23)

where $V_{\lambda }$ is determined in (22), and admits a unique solution for any $\rho \in (0,{\tilde{\rho }}_{\ast }]$.

Following the above result, in the general setting, the EWFMA chart elements are obtained through the scheme

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{\theta }}_j=arg\underset{\theta \in \tilde{\mathsf{\Theta }}}{min}\left\{\lambda \parallel \widehat{f}\left(\theta \right)-{\widehat{f}}_j{\parallel }_2^2+(1-\lambda ){\parallel \widehat{f}\left(\theta \right)-{\tilde{f}}_{j-1}\parallel }_2^2+R_{\rho }\left(\theta \right)\right\},\hfill \\ {\tilde{\theta }}_0={({\tilde{\alpha }}_0,{\tilde{\beta }}_0,{\tilde{\kappa }}_0,{\tilde{\zeta }}_0)}^{\prime }={(1,0,1,0)}^{\prime }\hfill \end{array}\right.\end{array}$$

(24)

Determining the optimal parameters ${\tilde{\theta }}_j$ is equivalent to the determination of ${\tilde{f}}_j$ in problem (21) under the shape-invariant model framework. Then, the modeled profile chart elements are obtained in a sense that is compatible with the space $\mathcal{M}$ through the notion of the Fréchet mean. The relevant chart for monitoring the modeled profiles’ deviance process (or simply the deformations’ deviance process) is constructed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tilde{D}}_j^{\theta }=\lambda D_j^{\theta }+(1-\lambda ){\tilde{D}}_{j-1}^{\theta },\lambda \in (0,1],j=1,2,\dots \hfill \\ {\tilde{D}}_0^{\theta }=0\hfill \end{array}\right.\end{array}$$

(25)

where $D_j^{\theta }:={\parallel \widehat{f}\left({\theta }_j\right)-f_0\parallel }_2^2$ with control limits of the chart to be specified by the induced empirical distribution from the training set.

Remark 3.

One may consider beginning the shift detection procedure by building typical EWMA charts on the obtained deformation parameters and then extending this chart for the deformation processes and the object $f_j$. However, the nonlinearity of $f_j$ with respect to the parameters (particularly for the shape-invariant model with respect to the phase deformation parameters) does not allow for such a generalization. Notice that, assuming that the deformation functions $\mathsf{\Lambda },\mathsf{\Pi }$ are linear with respect to their parameters, e.g., $\mathsf{\Lambda }\left(z\right)=\beta +\alpha z$, $\mathsf{\Pi }\left(s\right)=\zeta +\kappa s$, although one may attempt to monitor the deformation functions separately, e.g., ${\tilde{\mathsf{\Lambda }}}_j=\lambda {\mathsf{\Lambda }}_j+(1-\lambda ){\tilde{\mathsf{\Lambda }}}_{j-1}$ and $\tilde{\mathsf{\Pi }}j=\lambda {\mathsf{\Pi }}_j+(1-\lambda ){\tilde{\mathsf{\Pi }}}_{j-1}$, then the resulting chart for $f_j$ through ${\tilde{f}}_j=\lambda {\tilde{\mathsf{\Lambda }}}_j\circ f_0\circ \tilde{\mathsf{\Pi }}j+(1-\lambda ){\tilde{f}}_{j-1}$ does not coincide with the minimizer of Equation (21).

4.3. The Functional Profiles’ Shift Detection Algorithm

Combining the two shift detection schemes described in (20) and (25) results in an effective tool for both detecting and explaining potential shifts for profiles which can be captured by the parameterization offered by the shape-invariant model. At the first step, the observed profile’s underlying shape is checked, if it is sufficiently close to the estimated typical pattern ($f_0$). If a shift is detected at this step, then the profile is considered as an out of control (OOC) one, at least under the shape-invariant model’s explainability. If no shift is detected, then the shape-invariant model can be considered a valid approach and the scheme proceeds with studying the profile’s amplitude-phase deformation characteristics. At this step, the second monitoring approach is applied and determines whether the aggregate deformation effect leads to an IC status for the examined profile or not. If no shift is detected, the profile is considered as an IC one, while in the opposite case, a further investigation is performed on which deformation attributes led to the shift by constructing typical EWMA charts on each one of the deformation parameter in ${\theta }_j$. A graphical representation of the monitoring procedure is illustrated in Figure 2 while the steps of the monitoring scheme are briefly described in Algorithm 2.

Algorithm 2: The EWFMA shifts detection scheme

Phase I: Mean profile and IC profiles’ characteristics determination (a)  Mean pattern estimation Given a set of IC profiles $\mathcal{F}=\{f_1,f_2,\dots ,f_n\}$estimate the mean profile $f_0$according to the procedure described in Section 3.2. (b)  Estimation of the IC profiles deformation characteristics Determine the deformation characteristics of all IC profiles with respect to  $f_0$following the procedure described in Section 3.4 Phase II: Detection of shape and/or deformation shifts (a)  Shift detection on the underlying shape Given  $\lambda >0$, for a newly sampled profile  $f_j$, apply the procedure described in (20) and determine whether the underlying shape of the profile  $f_j$  is significantly shifted with regard to the reference mean shape  $f_0$. If no shift is detected, proceed to Step II(b). (b)  Shift detection on the deformation process Apply the chart described in (25) to determine whether the aggregate amplitude-phase deformation is an IC one. If a shift is detected, check the individual deformation characteristics (deformation parameters) through individual EWMA charts for the parameters obtained by (24).

4.4. Analysis of Ambient Air Quality Profiles in the Area of Athens

In this section, the presented methodology is implemented to analyze air pollution profiles in an area of the city of Athens. The discussed approach has been also previously tested with success through a number of simulation experiments with synthetic types of data. Since all the important aspects and capabilities of the method that have been displayed in the synthetic data experiments are fully observed through the air pollution case study without losing any part of the stress assessment of the approach, and taking into account (a) the better insights provided by a real-world example and (b) with the objective of keeping this paper concise, only the real-world example is illustrated. The functional modelling and shift detection approach presented in Section 3 is implemented in studying daily air pollution profiles in which measurements have been sampled by atmospheric pollution sensors from the area of Patission Street in Athens.

The task of monitoring ambient air quality in this area for the time period 2001–2007 under the functional data setting is performed. The available data (publicly available at: https://ypen.gov.gr/perivallon/poiotita-tis-atmosfairas/dedomena-metriseon-atmosfairikis-rypansis/ accessed on 12 August 2023) are provided by pollution sensors in the area which were installed by the Hellenic Ministry of Environment and Energy and consist of hourly measurements (mean values per hour for the duration of a day, i.e., 24 measurements) of the concentrations of four chemical quantities considered potential pollutants, and specifically, carbon monoxide (CO), nitrogen dioxide (NO${}_2$), ozone (O${}_3$), and sulfur dioxide (SO${}_2$). Certain safety concentration thresholds for human health concerning the concentrations of these pollutants in the regional atmosphere have been set by the recent directions of the World Health Organization (https://www.who.int/news-room/fact-sheets/detail/ambient-(outdoor)-air-quality-and-health accessed on 22 September 2023) (WHO) (please see

Table 1. Safety concentration thresholds of the pollutants under study for human health according to the World Health Organization.

Pollutant	Time Interval	Unit	Maximum AverageConcentration Threshold
Carbon monoxide (CO)	1 h	mg/m${}^3$	≤35
	8 h	mg/m${}^3$	≤10
	24 h	mg/m${}^3$	≤4
Nitrogen dioxide (NO${}_2$)	1 h	$\mathsf{\mu }$g/m${}^3$	≤200
	24 h	$\mathsf{\mu }$g/m${}^3$	≤25
Ozone (O${}_3$)	8 h	$\mathsf{\mu }$g/m${}^3$	≤60
Sulfur dioxide (SO${}_2)$	24 h	$\mathsf{\mu }$g/m${}^3$	≤40

). According to these guidelines, a day in which at least one of the thresholds was violated is characterized as an OOC day, while if none were violated, it is considered an IC day. This classification strategy is adopted in the experiment for distinguishing between IC and OOC profiles. However, note that these thresholds are constantly revised by the WHO with the tendency to become more strict over the years.

This threshold-based approach is maybe not the best strategy for the efficient and successful monitoring of ambient air quality, since the actual nature of data is of functional form while the threshold is defined pointwise (with respect to time and space). From this perspective, it could be possible that a pollutant’s concentration is in quite high levels during the day but never exceeds the nominal threshold value. Therefore, such a profile, although not considered an OOC one, would provide a significant risk to public health. The discussed functional approach provides an alternative tool for assessing and measuring the status of such phenomena that evolve simultaneously with a similar manner (with small fluctuations around a specific standard). Employing the framework of deformation models in the profile modelling task, and in particular, exploiting the explanatory capabilities of the shape-invariant model, allows for the recovery and quantification of special characteristics that cannot be captured by typical pointwise approaches. To illustrate the capabilities of the discussed method, the recorded profiles from the time period 2001–2004 are used as the training dataset, while the records from the time period 2005–2007 are used as the test dataset. Note that the focus in the analysis refers to the months October, November, and December, since this period is considered the “peak period” for the concentration of pollutants in the atmosphere (i.e., it is more interesting in terms of identifying the OOC days). Moreover, in these three months, no significant differences in the environmental conditions are observed and the median levels of the polluters are quite the same (similar median estimated curves) and as a result, the seasonality effect is not an issue.

During phase I of the monitoring procedure, the IC observations from the time period 2001–2004 are used as the trainset and the typical behavior for the intra-day concentration of each pollutant (i.e., the typical curve represents the daily evolution of each pollutant) alongside the respective deformations around the mean patterns and their induced empirical distributions are estimated. A slightly modified version of the shape-invariant model is incorporated where the phase scaling parameter $\kappa$ is omitted to simplify the analysis. The deformation characteristics parameterized by the SIM (location, amplitude and phase) allow for a more careful and meaningful representation of the potential deviations from the reference pattern that could possibly affect the pollutant’s status each day. Each one of the considered pollutants display quite different concentration behaviors which enable the assessment of the presented methodology under different conditions and data patterns. In Figure 3, the IC daily concentration curves of the pollutants under study (recorded in the training set) and their estimated mean patterns (typical concentration profiles) are illustrated. All data patterns seems to be successfully calibrated by the shape-invariant model and by the calculated Fréchet means, plausibly representing the typical behaviors of the pollutants’ concentrations in the city’s regional atmosphere.

Subsequently, phase II of the functional monitoring scheme is performed and the method is assessed over the test dataset concerning the time period 2005–2007. Following the described procedure, the EWFMA control charts for the sampled profiles are constructed to detect significant shifts either in shape and/or in the deformation process. The resulting diagnostic charts are depicted for each pollutant under study in Figure 4, Figure 5, Figure 6 and Figure 7. In the upper panel, the deviance-related charts are displayed with respect to the reference shape (left column) and the total deformation effect with respect to the estimated mean pattern (right column). The second panel of charts concerns shifts with respect to the individual deformation features explained by the shape-invariant model (shape location, shape amplitude, and phase shift). Note that the Fréchet mean is calculated according to the $L^2$-metric sense, while exponential weighting parameters on the grid

$$\lambda =0.01,0.02,\dots ,0.05,0.10,\dots ,0.90,0.95,0.96,\dots ,0.99$$

have been tested. The optimal choice was selected per case (the term optimality refers to the choice of $\lambda$ for which less OOC observations are classified as IC and less IC observations are classified as OOC, i.e., minimizing both type-I and type-II errors). Although one could try to occasionally reallocate the weighting parameter value by separating the train set to further parts and re-arranging the values according into cross-validation findings, such a task is beyond the scopes of this work and therefore a constant value for each pollutant for the whole monitoring task is used.

In

Table 2. Per stage performance of the functional EWMA-type monitoring scheme for each pollutant.

	Monitoring	Number	Classification Accuracy		Within Class Accuracy
Pollutant	Process	of Cases	True (%)	False (%)	OOC (%)	IC (%)
	Both	266	87.22 (232)	12.78 (34)	100.00 (71)	82.56 (161)
CO	Shape process	266	69.55 (185)	30.45 (81)	2.82$$ (2)	93.85 (183)
	Deformation process	266	91.35 (243)	8.65 (23)	100.00 (71)	88.21 (172)
	Both	264	82.58 (218)	17.42 (46)	78.26 (18)	82.99 (200)
NO${}_2$	Shape process	264	81.82 (216)	18.18 (48)	0.00$$ (0)	89.63 (216)
	Deformation process	264	84.85 (224)	15.15 (40)	78.26 (18)	85.48 (206)
	Both	266	88.35 (235)	11.65 (31)	100.00 (1)	88.30 (234)
O${}_3$	Shape process	266	96.99 (258)	3.01$$ (8)	0.00 (0)	97.36 (258)
	Deformation process	266	90.23 (240)	9.77 (26)	100.00 (1)	90.19 (239)
	Both	266	95.86 (255)	4.14 (11)	100.00 (55)	94.79 (200)
SO${}_2$	Shape process	266	77.44 (206)	22.56 (60)	0.00$$ (0)	97.63 (206)
	Deformation process	266	96.62 (257)	3.38$$ (9)	100.00 (55)	95.73 (202)

, the performance diagnostics for each monitoring stage of the method are illustrated, namely (a) the shape process; (b) the deformation process; and (c) the profile in total, relying on the modelling capabilities and explainability of the shape-invariant model. The classification accuracy of the scheme is assessed both in total and within each profile category (IC and OOC ones) through the relevant true classification and wrong classification percentages. The classification accuracy section of the table refers to the total true classification percentages, i.e., the IC and OOC profiles that are correctly identified by the scheme, and false classification percentages, i.e., cases of IC profiles that are identified as OOC and the opposite. The within class accuracy section, refers to the percentages of correct identifications within each class, i.e., the OOC column depicts the percentage of OOC cases that are correctly identified as OOC comparing to the total number of OOC cases, while the IC column depicts the percentage of IC cases that are correctly identified as IC compared to the total number of the IC cases. The shift detection scheme displayed in general a good performance in identifying the shifts (according to the threshold definition of WHO discussed above) with true classification percentages between 82 and 96%. Clearly, the pollutants CO and NO${}_2$ can be characterized as the more difficult cases where higher error rates are observed, while the SO${}_2$ is the pollutant where the scheme displayed the best accuracy. From the results, concerning the underlying shape process monitoring (rows about the shape process in Table 2 and the first deviance plot in the upper panel of the individual figures per pollutant), it is evident that the SIM is appropriate for describing such a type of profile, since too few cases are indicated as not explainable by this model (OOC cases). Note that, in the table, the results were calculated according to the classification by WHO and not according to actual shape shift identification. The deformation process shift detection scheme also performed decently in identifying actual shifts for the pollutants CO, O${}_3$, and SO${}_2$ with the total true classification percentage between 90 and 96% and zero probabilities in conducting type-I error. However, in the case of NO${}_2$, the error rates are much higher (around 15–20%). The individual deformation characteristics can be further investigated through the individual deformation parameter charts when a shift is detected. For all pollutants, the phase shift attribute quantified by the parameter $\zeta$ does not seem to be related to the actual shifts, and in the majority of cases, the related charts do not display behaviors beyond the limits. On the contrary, the phase amplitude parameter $\alpha$ and in some cases the vertical shift parameter $\beta$ identify the significant changes (shape deformations) in the daily concentration profiles as displayed from the relevant plots (see the second panel of graphs in Figure 4, Figure 5, Figure 6 and Figure 7). In general, it seems that the amplitude deformation level can be used as a proxy for detecting significant shifts when the shape process is in control.

4.5. Methodological Overview

The presented methodology provides a complete framework for both tasks of statistical modelling and the monitoring of functional profiles, building on the combination of the general applicability of the Fréchet mean (FM), and the explanatory capabilities of the shape-invariant model (SIM). The obtained methodological contributions are summarized below:

• A compatible version of the FM with the SIM is determined in (6), expressing the typical profile in a semi-parametric form that depends both on (a) the sample of the functional profiles and (b) the deformation features incorporated by SIM. The typical profile (i.e., the respective Fréchet mean of the sampled profiles) is determined by the solution of problem (7). Since the optimal solution of the problem depends on the nature of the sampled profiles, the regularized version of the problem stated in (10) is studied to provide a more stable estimation procedure, while the relevant numerical approximation scheme is described in Algorithm 1.
• The optimal deformation features of the profiles under study with respect to the typical one (estimated on the previous step) are obtained through the solution of the profile registration problem (17) for each one of the profiles. This results in a dataset of deformation parameters that allows one to specify the deviance in each profile from the mean pattern with respect to the different deformation features that are taken into account by the SIM.
• A modified version of the well-known EWMA monitoring scheme is developed that detects the potential shifts of newly sampled profiles either on shape, or on the deformation features (or both). The SIM-compatible version of the Fréchet mean and the optimal deformation features of the profiles in the training set obtained in the above steps are employed to characterize the IC behavior. Potential shifts in the underlying shape of the newly sampled profile are detected by the scheme stated in (20), while potential shifts in its deformation comparing to the typical shape are detected by the scheme stated in (25) which relies on the solution of the optimization problem (22). These steps are performed successively to characterize a newly sampled profile as an IC one or as an OOC one, according to the stages of Algorithm 2 and the illustration provided in Figure 2.

5. Conclusions

In this work, an appropriate framework for the statistical treatment and monitoring of functional profiles is presented. Relying on the notion of the Fréchet mean (which offers a generalized sense of the mean for metric space valued data, such as curves), combined with the framework of the SIM, a modelling procedure has been proposed that allows both for (a) the characterization of the typical (mean) profile and (b) the description of acceptable deformations/deviations from the typical profile in terms of the deformation features that are incorporated by the SIM (i.e., amplitude and phase deformations). At the next step, the obtained typical pattern (i.e., the obtained Fréchet mean of the set of profiles under study) and the induced sample of deformation parameters and their distributions are used to develop a monitoring scheme compatible with the functional data settings. The proposed scheme is an extension, for functional data, of the well-known EWMA monitoring process, employing the notion of the Fréchet mean combined with the shape-invariant model (SIM). The scheme proceeds along two sequential stages. First, potential shifts (deformations) of the underlying shape of the profile under study are examined as compared to the typical shape. If a shift is detected at this stage, the sampled profile is considered an OOC one (shape shift). In case no shift is detected in the shape process, the second stage of the scheme examines whether the deformations of the profile lie within the acceptable standards of deviations from the typical behavior. Potential shifts in the deformation process can be further allocated to certain deformation features like amplitude deformations, phase deformations, etc.

For the proper illustration and assessment of the presented methodology, air pollution profiles for pollutants from an area in the city of Athens were studied. The described modelling framework has been implemented to capture the typical intra-day patterns of the pollutants and the relevant distribution of the deformation features that characterize the acceptable deviance from the IC behavior. Next, the developed monitoring scheme was implemented for detecting shifts in the daily sampled profiles of the pollutants with success. In particular, the total accuracy in correctly identifying the status of the profiles varies between 83% and 96%. For the IC category, the percentage of true identification varies between 83% and 95%. Moreover, concerning the OOC category, for the three pollutants, all the OOC profiles were successfully identified with the exception of one pollutant where the percentage of true classification was about 78%. Although the percentages of correct identification are quite high in most cases, the monitoring scheme seems to present a more conservative behavior when characterizing profiles as IC leading to higher Type II errors, while Type I errors are almost zero. The higher Type II errors in these cases could be reduced by including more information in the monitoring task (e.g., take into account extra environmental quantities like temperature, humidity, etc.), or by the more careful modelling of the interdependencies concerning the deformation features within and across the different pollutants, or even by considering a different type of deformation model to SIM in case that the pollutant being studied provides quite different patterns which should still be considered as IC ones (e.g., by using landmark deformation models [33]). In any case, all the above directions can be considered extensions of the current framework and potential steps for future research.